[PPT] - I n fj n i t e - n o i s e c r i t i c a l i t PowerPoint Presentation

SLIDE 1

I n fj n i t e

n
i

s e c r i t i c a l i t y

J

s

é A . H

y
s

* I n s t i t u t

d

e F í s i c a d e S ã

C

a r l

s
U

S P

C

l

l a b

r

a t i

n

: T h

ma

s V

j

t a * h

y
s

@ i f s c . u s p . b r Financial support:

E P L 1 1 2 , 3 2 ( 2 1 5 ) a r X i v : 1 5 7 . 5 6 7 7

SLIDE 2

O u t l i n e

I

n t r

d

u c t i

n

: t h e c

n

t a c t p r

c

e s s

– D

e fj n i t i

n

a n d a p p l i c a t i

n

s t

n
n

e q u i l i b r i u m P T

C

r i t i c a l i t y

n

a fm u c t u a t i n g ( n

i

s y ) e n v i r

n

me n t

– I

n fj n i t e

n
i

s e c r i t i c a l i t y a t d=1: r a n d

m

w a l k

– I

n fj n i t e

n
i

s e c r i t i c a l i t y a t fj n i t e d: r e a l

t

i me R G

O

b s e r v a b l e s a n d c r i t i c a l b e h a v i

u

r

C
n

c l u s i

n

s

SLIDE 3

T h e c

n

t a c t p r

c

e s s : d e fj n i t i

n

S t

c

h a s t i c d y n a mi c s : D e c a y ( h e a l i n g ) r a t e : O fg s p r i n g p r

d

u c t i

n

( i n f e c t i

n

) r a t e : Prototypical model for epidemic spreading

T. E. Harris, Ann. Prob. 2, 969 (1974)

A C T I V E ( i n f e c t e d ) I N A C T I V E ( h e a l t h y ) A c t i v e s i t e s s t e a d y

s

t a t e d e n s i t y :

SLIDE 4

T h e c

n

t a c t p r

c

e s s : n

n

e q . P T

N E P T i n t

a

n a b s

r

b i n g s t a t e a t D i r e c t e d P e r c

l

a t i

n

u n i v e r s a l i t y c l a s s A c t i v e ( fm u c t u a t i n g ) s t e a d y s t a t e A b s

r

b i n g ( i n a c t i v e ) s t a t e A c t i v e s i t e s s t e a d y

s

t a t e d e n s i t y :

SLIDE 5

T h e c

n

t a c t p r

c

e s s : D P u n i v e r s a l i t y c l a s s

P e r c

l

a t i

n

a n d d i r e c t e d p e r c

l

a t i

n

: g e

me

t r i c P T s T h e C P c a n b e ma p p e d

n

a d+1 D P g e

me

t r i c p r

b

l e m 2 n d

r

d e r P T

P i c t u r e c r e d i t : H i n r i c h s e n , A d v . P h y s . 4 9 , 8 1 5 ( 2 )

SLIDE 6

O u r mo d e l

C

n

t a c t p r

c

e s s i n a fm u c t u a t i n g ( n

i

s y ) e n v i r

n

me n t :

H e a l i n g r a t e : I n f e c t i

n

r a t e : t i me T i me i n t e r v a l s a r e fj n i t e . T i me

d

e p e n d e n t i i d

SLIDE 7

Wh a t d

w

e w a n t t

k

n

w

?

C

n

t a c t p r

c

e s s i n a fm u c t u a t i n g ( n

i

s y ) e n v i r

n

me n t :

H e a l i n g r a t e : I n f e c t i

n

r a t e : t i me D e n s i t y P D F a s a f u n c t i

n
f

t i me ? I s t h e me a n d e n s i t y r e p r e s e n t a t i v e

f

t h e ma n y p

s

s i b l e t i me h i s t

r

i e s ? L i f e t i me

f

fj n i t e s y s t e ms ? C

r

r e l a t i

n

l e n g t h / t i me ? T i me

d

e p e n d e n t i i d

SLIDE 8

O u r mo d e l : l i mi t i n g c a s e s

C

n

t a c t p r

c

e s s i n a fm u c t u a t i n g ( n

i

s y ) e n v i r

n

me n t :

H e a l i n g r a t e : I n f e c t i

n

r a t e : t i me I n a l l t i me i n t e r v a l s , t h e s y s t e m i s i n t h e A C T I V E p h a s e : A B S O R B I N G p h a s e : T h e n , u s u a l d y n a mi c s . T i me d i s

r

d e r h a s “ l i t t l e ” e fg e c t s . I n t e r e s t i n g c a s e : “ t i me p a t c h e s ” a r e i n

p

p

s

i t e p h a s e s – p h a s e t r a n s i t i

n

. T i me

d

e p e n d e n t i i d

SLIDE 9

Wh a t d

w

e e x p e c t ?

L a r g e d e n s i t y fm u c t u a t i

n

s – fj e l d

t

h e

r

e t i c a l me t h

d

s a r e n

t

a p p r

p

r i a t e . Mo n t e C a r l

t

i me e v

l

u t i

n
f

t h e d e n s i t y a t c r i t i c a l i t y .

SLIDE 10

Me a n fj e l d a p p r

a

c h

H

w

d

w

e f a c e t h e p r

b

l e m? L e t ' s t r y s

me

t h i n g s i mp l e . L a n g e v i n E q . i n MF l e v e l : I n t e g r a t i n g : N

n

l i n e a r t e r m p r e v e n t i n g ½>1 L e t s n e g l e c t i t a n d r e c

v

e r l i n e a r i t y L e t R A N D O M WA L K Wh a t i s t h e r

l

e p l a y e d b y ?

T
e

n s u r e t h e w a l k e r i s i n t h e p

s

i t i v e s i d e H

w

d

w

e p u t t h e n

n

l i n e a r i t y b a c k i n t h e g a me ? R A N D O M WA L K w i t h a r e fm e c t i n g h a r d w a l l

SLIDE 11

Me a n fj e l d a p p r

a

c h : P h a s e t r a n s i t i

n

P r

b

.

f

fj n d i n g t h e w a l k e r b e t w e e n [ x,x+dx] a t t i me R A N D O M WA L K w i t h a r e fm e c t i n g h a r d w a l l S

l

v e H a r d w a l l c

n

s t r a i n t a t x=0.

SLIDE 12

Me a n fj e l d a p p r

a

c h : P h a s e t r a n s i t i

n

P r

b

.

f

fj n d i n g t h e w a l k e r b e t w e e n [ x,x+dx] a t t i me R A N D O M WA L K w i t h a r e fm e c t i n g h a r d w a l l 1

s t

r e g i me : I n a c t i v e p h a s e U s u a l G a u s s i a n I mp l i c a t i

n

s :

SLIDE 13

Me a n fj e l d a p p r

a

c h : P h a s e t r a n s i t i

n

P r

b

.

f

fj n d i n g t h e w a l k e r b e t w e e n [ x,x+dx] a t t i me R A N D O M WA L K w i t h a r e fm e c t i n g h a r d w a l l 2

n d

r e g i me : A c t i v e p h a s e S t a t i

n

a r y d i s t r i b u t i

n

I mp l i c a t i

n

s : S t r

n

g d e n s i t y fm u c t u a t i

n

s n e a r c r i t i c a l i t y v!0

SLIDE 14

Me a n fj e l d a p p r

a

c h : P h a s e t r a n s i t i

n

P r

b

.

f

fj n d i n g t h e w a l k e r b e t w e e n [ x,x+dx] a t t i me R A N D O M WA L K w i t h a r e fm e c t i n g h a r d w a l l 3

r d

r e g i me : C r i t i c a l p

i

n t H a l f G a u s s i a n I mp l i c a t i

n

s : I n fj n i t y d e n s i t y fm u c t u a t i

n

s

I n fj n i t e

n
i

s e c r i t i c a l i t y

SLIDE 15

Me a n fj e l d : c

mp

a r i s

n

w i t h n u me r i c s

H a l f G a u s s i a n a t c r i t i c a l i t y : i n a g r e e me n t w i t h

u

r e x p e c t a t i

n

s

SLIDE 16

Me a n fj e l d a p p r

a

c h : L i f e t i me

A n y fj n i t e s y s t e m e v e n t u a l l y d e c a y s i n t

t

h e a b s

r

b i n g s t a t e 1

s t

p a s s a g e t i me A c t i v e p h a s e C r i t i c a l i t y I n a c t i v e p h a s e

SLIDE 17

Me a n fj e l d a p p r

a

c h : Wh a t a r e w e d

i

n g ?

F r

m

c

mp

l i c a t e d L a n g e v i n t

s

i mp l e l

g

i s t i c e q u a t i

n

1 . T h i s l

g

i s t i c e q u a t i

n

i s a l r e a d y i n t e r e s t i n g

n

i t s

w

n . I t mo d e l s p r

c

e s s e s i n w h i c h d i fg u s i v e mi x i n g i s s t r

n

g e n

u

g h t

w

i p e

u

t s p a t i a l s t r u c t u r e s

B

i

l
g

i c a l p

p

u l a t i

n

, c h e mi c a l r e a c t i

n

s , l i n g u i s t i c s , e t c . . . 2 . F

r

t h e C P a t fj n i t e d i me n s i

n

s , s p a t i a l c

r

r e l a t i

n

s a r e i mp

r

t a n t .

H
w

d

w

e a c c

u

n t f

r

t h e m?

R

e a l

t

i me R G D

e

s t h e i n fj n i t e

n
i

s e c r i t i c a l i t y s u r v i v e a t fj n i t e d?

SLIDE 18

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 19

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 20

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 21

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 22

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 23

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 24

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s

SLIDE 25

R e a l

t

i me R G

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s H

w

d

w

e d

i

t ? E l i mi n a t e t h e s ma l l e s t u p w a r d a n d d

w

n w a r d s e g me n t s I t e r a t i v e ma p : n

t

mu c h i mp

r

t a n t f

r

s ma l l d e n s i t i e s

U p w a r d s e g me n t : D

w

n w a r d s e g me n t : C u t

fg

:

SLIDE 26

R e a l

t

i me R G : d e c i ma t i

n

p r

c

e d u r e

E x p

n

e n t i a l d e c a y

f

a c t i v i t y

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s H

w

d

w

e d

i

t ? E l i mi n a t e t h e s ma l l e s t u p w a r d a n d d

w

n w a r d s e g me n t s I t e r a t i v e ma p : n

t

mu c h i mp

r

t a n t f

r

s ma l l d e n s i t i e s

U p w a r d s e g me n t : D

w

n w a r d s e g me n t : C u t

fg

:

SLIDE 27

R e a l

t

i me R G : d e c i ma t i

n

p r

c

e d u r e

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s H

w

d

w

e d

i

t ? E l i mi n a t e t h e s ma l l e s t u p w a r d a n d d

w

n w a r d s e g me n t s I t e r a t i v e ma p : n

t

mu c h i mp

r

t a n t f

r

s ma l l d e n s i t i e s

U p w a r d s e g me n t : D

w

n w a r d s e g me n t : C u t

fg

:

SLIDE 28

Me a n fj e l d : e x p

n

e n t i a l a c t i v i t y s p r e a d i n g F i n i t e d: Ma x i mu m s p r e a d i n g : b a l l i s t i c

R e a l

t

i me R G : d e c i ma t i

n

p r

c

e d u r e

I d e a : c

a

r s e

g

r a i n s ma l l t i me s t r u c t u r e s H

w

d

w

e d

i

t ? E l i mi n a t e t h e s ma l l e s t u p w a r d a n d d

w

n w a r d s e g me n t s I t e r a t i v e ma p : n

t

mu c h i mp

r

t a n t f

r

s ma l l d e n s i t i e s

U p w a r d s e g me n t : D

w

n w a r d s e g me n t : C u t

fg

:

SLIDE 29

R e a l

t

i me R G : t h e R G fm

w

I d e a : F

l

l

w

t h e d i s t r i b u t i

n

s

f

a n d a s V a r i a b l e c h a n g i n g :

F l

w

e q u a t i

n

s : F i x e d p

i

n t s

l

u t i

n

s : “ K

r

s t e r l i t z

T

h

u

l e s s ” R G fm

w

SLIDE 30

R e a l

t

i me R G : t h e R G fm

w

“ K

r

s t e r l i t z

T

h

u

l e s s ” R G fm

w

R G t r a j e c t

r

i e s :

SLIDE 31

R e a l

t

i me R G : t h e p h a s e s

I N A C T I V E p h a s e A C T I V E p h a s e C r i t i c a l i t y

I n fj n i t e

n
i

s e c r i t i c a l i t y

SLIDE 32

R e a l

t

i me R G :

b

s e r v a b l e s

K e e p t r a c k

f

a n d t h e t i me i n t e r v a l s R e c a l l t h a t L I F E T I ME = t i me i n t e r v a l w h e n I n a c t i v e P h a s e : A c t i v e P h a s e : C r i t i c a l i t y :

SLIDE 33

R e a l

t

i me R G : c

mp

a r i s

n

w i t h n u me r i c s

C r i t i c a l i t y d=2

I n a g r e e me n t w i t h

u

r p r e d i c t i

n

S y s t e m s i z e

SLIDE 34

C

n

c l u s i

n

s

I n fj n i t e

n
i

s e c r i t i c a l i t y : E n v i r

n

me n t a l n

i

s e ( t i me fm u c t u a t i

n

s )

n

t h e p a r a me t e r s

f

t h e C

n

t a c t P r

c

e s s y i e l d s t

a

n

n
e

q u i l i b r i u m P T c h a r a c t e r i z e d b y s t r

n

g d e n s i t y fm u c t u a t i

n

s w i t h r e s p e c t t

t

h e ma n y d i fg e r e n t h i s t

r

i e s .

E P L 1 1 2 , 3 2 ( 2 1 5 ) a r X i v : 1 5 7 . 5 6 7 7

SLIDE 35

R e l e v a n c e

f

d i s

r

d e r : H a r r i s c r i t e r i

n
N

e c e s s a r y c

n

d i t i

n

f

r

t h e s t a b i l i t y

f

t h e c l e a n fj x e d p

i

n t

B

a s e d

n

t h e a v e r a g e b e h a v i

r
f

d i s

r

d e r

H a r r i s c r i t e r i

n

:

A. B. Harris, J. Phys. C 7, 1671 (1974)

» ...

Transition

Broadens or sharpens?

±r r

SLIDE 36

T h e u b i q u i t

u

s D P u n i v e r s a l i t y c l a s s

I mp

r

t a n t c

n

j e c t u r e : a l l c

n

t i n u

u

s N E P T s i n t

a

s i n g l e e fg e c t i v e a b s

r

b i n g s t a t e i n t h e a b s e n c e

f

a n y e x t r a s y mme t r y , l

n

g

r

a n g e p r

c

e s s e s ,

r

c

n

s e r v a t i

n

l a w i s i n t h e D P u n i v e r s a l i t y c l a s s .

J

a n s s e n ' 8 1 + G r a s s b e r g e r ' 8 2

I mp l i c a t i

n

s : O n e c a n e x t e n d t h e r u l e s

f

t h e g a me S e l f

d

e s t r u c t i

n

O fg s p r i n g p r

d

u c t i

n

D i fg u s i

n

C

a

g u l a t i

n

C P i s t h e I s i n g mo d e l

f

t h e a b s

r

b i n g

s

t a t e N E P T s H

w

e v e r , a l t h

u

g h t h e c r i t i c a l e x p

n

e n t s

f

C P a r e k n

w

n n u me r i c a l l y ( Mo n t e C a r l

a

mo n g

t

h e r me t h

d

s ) , t h e r e i s n

a

n a l y t i c a l s

l

u t i

n

f

r

t h i s mo d e l . . . F

r

b i d d e n p r

c

e s s

SLIDE 37

D P u n i v e r s a l i t y c l a s s : a n a l y t i c a l t

l

b

x

D P L a n g e v i n e q u a t i

n

( ma s t e r e q u a t i

n

)

G a u s s i a n n

i

s e :

R e g g e

n

fj e l d t h e

r

y ( a f t e r i n t e g r a t i n g

u

t ³)

Ma r t i n

S

i g g i a

R
s

e r e s p

n

s e fj e l d F r

m

h e r e , G r e e n ' s f u n c t i

n

a n d F e y n ma n d i a g r a ms

H a mi l t

n

i a n f

r

mu l a t i

n

( H i s a g e n e r a t

r
f

a Ma r k

v

p r

c

e s s )

i s t h e p r

b

.

f

fj n d i n g t h e s y s t e m i n s t a t e a t t i me t F

r

t h e C P : S t e a d y s t a t e – g r

u

n d s t a t e

f

H L

n

g

t

i me r e l a x a t i

n

– l

w
e

n e r g y s p e c t r u m

f

H P r i c e t

p

a y : n

n
h

e r mi t e a n ma t r i c e s

SLIDE 38

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

E x p e r i me n t a l s e t u p

D e s c r i p t i

n

:

N

e ma t i c l i q u i d c r y s t a l b e t w e e n t w

t

r a n s p a r e n t e l e c t r

d

e s .

C

a r r

H

e l f r i c h i n s t a b i l i t y f

r

s u ffjc i e n t l y h i g h V ( a n d l

w

!) a t u r b u l e n t fm

w

.

F
r

h i g h e r V, n e w t u r b u l e n t r e g i me s e x h i b i t i n g s p a t i

t

e mp

r

a l l y i n t e r mi t t e n t d y n a mi c s .

T

w

t
p
l
g

i c a l l y d i s t i n c t t u r b u l e n t r e g i me s : D S M1 a n d D S M2 ( w i t h d i s c l i n a t i

n

d e fg e c t s ) .

D

S M2 c a n d i s a p p e a r i n t

D

S M1 a n d i n d u c e t h e c r e a t i

n
f

n e w D S M2 ( l i k e a 2 + 1

d

C P ) .

1 4 mm

2

O b s e r v a t i

n

a l w i n d

w

1 2 1 7 x 9 1 1 ¹m

2

)

SLIDE 39

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

A c q u i r i n g t h e d a t a

SLIDE 40

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

Mo v i e

f

t h e t u r b u l e n t fm

w

: D S M1 ( D S M2 ) r e g i

n

s c

r

r e s p

n

d t

l

i g h t e r ( d a r k e r ) p a t c h e s .

SLIDE 41

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

T h e y c

n

fj r m 1 2 c r i t i c a l e x p

n

e n t s , 5 s c a l i n g f u n c t i

n

s , a n d 8 s c a l i n g r e l a t i

n

s

SLIDE 42

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

T h e y c

n

fj r m 1 2 c r i t i c a l e x p

n

e n t s , 5 s c a l i n g f u n c t i

n

s , a n d 8 s c a l i n g r e l a t i

n

s

SLIDE 43

D P a n d t u r b u l e n t l i q u i d c r y s t a l s

T a k e u c h i e t a l . , P R E 8 , 5 1 1 1 6 ( 2 9 )

T h e y c

n

fj r m 1 2 c r i t i c a l e x p

n

e n t s , 5 s c a l i n g f u n c t i

n

s , a n d 8 s c a l i n g r e l a t i

n

s

SLIDE 44

T h e u b i q u i t

u

s D P u n i v e r s a l i t y c l a s s ?

E x a mp l e s :

E

p i d e mi c s

C

e l l u l a r a u t

ma

t

n
C

a t a l y t i c c h e mi c a l r e a c t i

n

s

S

p a t i

t

e mp

r

a l i n t e r mi t t e n c y i n t u r b u l e n t fm u i d s

G

r

w

i n g i n t e r f a c e s , e t c .

H

. H i n r i c h s e n , A d v . P h y s . 4 9 , 8 1 5 ( 2 )

D

r i v e n p

l

y me r i c s u s p e n s i

n

s

D

y n a mi c s

f

S C v

r

t i c e s

Mu

t u a l i s m i n b a c t e r i a l c

l
n

i e s

A l t h

u

g h t h e D P u n i v e r s a l i t y c l a s s s e e ms t

b

e u b i q u i t

u

s … “ . . . t h e r e i s s t i l l n

e

x p e r i me n t w h e r e t h e c r i t i c a l b e h a v i

r
f

D P w a s s e e n . T h i s i s a v e r y s t r a n g e s i t u a t i

n

i n v i e w

f

t h e v a s t a n d s u c c e s s i v e t h e

r

e t i c a l e fg

r

t s ma d e t

u

n d e r s t a n d i t . D e s i g n i n g a n d p e r f

r

mi n g s u c h a n e x p e r i me n t h a s t h u s t

p

p r i

r

i t y i n m y l i s t

f
p

e n p r

b

l e ms ” .

P

. G r a s s b e r g e r , D i r e c t e d p e r c

l

a t i

n

: r e s u l t s a n d

p

e n p r

b

l e ms , i n N

n

l i n e a r i t i e s i n c

mp

l e x S y s t e ms , e d i t e d b y S . P u r i e t a l . ( 1 9 9 7 ) .

S

e e a l s

H

. H i n r i c h s e n , B r a z . J . P h y s . 3 , 6 9 ( 2 )

SLIDE 45

T h e u b i q u i t

u

s D P u n i v e r s a l i t y c l a s s ?

T h e c u l p r i t : D I S O R D E R H a r r i s c r i t e r i

n

: s p a t i a l q u e n c h e d d i s

r

d e r i s r e l e v a n t i f t e mp

r

a l d i s

r

d e r i s r e l e v a n t i f S p a t i a l q u e n c h e d d i s

r

d e r i s r e l e v a n t i n mo s t

f

t h e i n t e r e s t i n g c a s e s T e mp

r

a l d i s

r

d e r i s r e l e v a n t i n a l l c a s e s s p a t i a l q u e n c h e d d i s

r

d e r : r a t e s a r e t i me

i

n d e p e n d e n t r a n d

m

v a r i a b l e s . t e mp

r

a l d i s

r

d e r : r a t e s a r e s p a c e

i

n d e p e n d e n t r a n d

m